KoAT2 Proof Output

Initial Problem

Start: eval_noiseTickSize_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: nondef.0
Locations: eval_noiseTickSize_0, eval_noiseTickSize_1, eval_noiseTickSize_bb0_in, eval_noiseTickSize_bb1_in, eval_noiseTickSize_bb2_in, eval_noiseTickSize_start, eval_noiseTickSize_stop
Transitions:
t₄: eval_noiseTickSize_0(X₀, X₁, X₂, X₃) → eval_noiseTickSize_1(nondef.0, X₁, X₂, X₃)
t₅: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb1_in(X₀, X₁, 1+X₂, 1) :|: 1+X₀ ≤ 0 ∧ 2+X₂ ≤ X₁
t₆: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb1_in(X₀, X₁, 1+X₂, 1) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ 0 ∧ 2+X₂ ≤ X₁
t₇: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb1_in(X₀, X₁, 1+X₂, 1) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ 0 ∧ 2+X₂ ≤ X₁
t₈: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb1_in(X₀, X₁, 1+X₂, 1) :|: 1 ≤ X₀ ∧ 2+X₂ ≤ X₁
t₁₃: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb1_in(X₀, X₁, X₂, 1) :|: 1+X₀ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
t₁₄: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb1_in(X₀, X₁, X₂, 1) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
t₁₇: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb1_in(X₀, X₁, 1+X₂, 1+X₃) :|: X₃ ≤ 10000 ∧ 1+X₀ ≤ 0 ∧ 2+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
t₁₈: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb1_in(X₀, X₁, 1+X₂, 1+X₃) :|: X₃ ≤ 10000 ∧ 1 ≤ X₀ ∧ 2+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
t₂₃: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb1_in(X₀, X₁, X₂, 1+X₃) :|: X₃ ≤ 10000 ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
t₉: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb2_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 1+X₂ ∧ 1+X₀ ≤ 0
t₁₀: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb2_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 1+X₂ ∧ 1 ≤ X₀ ∧ 1+X₀ ≤ 0
t₁₁: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb2_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 1+X₂ ∧ 1 ≤ X₀ ∧ 1+X₀ ≤ 0
t₁₂: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb2_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 1+X₂ ∧ 1 ≤ X₀
t₁₅: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ X₁ ≤ X₂
t₁₆: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ X₁ ≤ X₂
t₁₉: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0 ∧ 10001 ≤ X₃ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
t₂₀: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb2_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 1+X₂ ∧ 1+X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
t₂₁: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 10001 ≤ X₃ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
t₂₂: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb2_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 1+X₂ ∧ 1 ≤ X₀ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
t₂₄: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb2_in(X₀, X₁, X₂, X₃) :|: 10001 ≤ X₃ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
t₂₅: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ X₁ ≤ X₂
t₁: eval_noiseTickSize_bb0_in(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb1_in(X₀, X₁, 0, 0)
t₂: eval_noiseTickSize_bb1_in(X₀, X₁, X₂, X₃) → eval_noiseTickSize_0(X₀, X₁, X₂, X₃)
t₂₆: eval_noiseTickSize_bb2_in(X₀, X₁, X₂, X₃) → eval_noiseTickSize_stop(X₀, X₁, X₂, X₃)
t₀: eval_noiseTickSize_start(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Cut unsatisfiable transition [t₆: eval_noiseTickSize_1→eval_noiseTickSize_bb1_in; t₇: eval_noiseTickSize_1→eval_noiseTickSize_bb1_in; t₁₀: eval_noiseTickSize_1→eval_noiseTickSize_bb2_in; t₁₁: eval_noiseTickSize_1→eval_noiseTickSize_bb2_in; t₁₃: eval_noiseTickSize_1→eval_noiseTickSize_bb1_in; t₁₄: eval_noiseTickSize_1→eval_noiseTickSize_bb1_in; t₁₅: eval_noiseTickSize_1→eval_noiseTickSize_bb2_in; t₁₆: eval_noiseTickSize_1→eval_noiseTickSize_bb2_in; t₁₇: eval_noiseTickSize_1→eval_noiseTickSize_bb1_in; t₁₈: eval_noiseTickSize_1→eval_noiseTickSize_bb1_in; t₁₉: eval_noiseTickSize_1→eval_noiseTickSize_bb2_in; t₂₀: eval_noiseTickSize_1→eval_noiseTickSize_bb2_in; t₂₁: eval_noiseTickSize_1→eval_noiseTickSize_bb2_in; t₂₂: eval_noiseTickSize_1→eval_noiseTickSize_bb2_in]

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₂ for location eval_noiseTickSize_bb2_in

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₂ for location eval_noiseTickSize_1

Found invariant X₃ ≤ 10001 ∧ X₃ ≤ 10001+X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₂ for location eval_noiseTickSize_bb1_in

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₂ for location eval_noiseTickSize_0

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₂ for location eval_noiseTickSize_stop

Problem after Preprocessing

MPRF for transition t₅: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb1_in(X₀, X₁, 1+X₂, 1) :|: 1+X₀ ≤ 0 ∧ 2+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

• eval_noiseTickSize_0: [X₁-1-X₂]
• eval_noiseTickSize_1: [X₁-1-X₂]
• eval_noiseTickSize_bb1_in: [X₁-1-X₂]

MPRF for transition t₈: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb1_in(X₀, X₁, 1+X₂, 1) :|: 1 ≤ X₀ ∧ 2+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

• eval_noiseTickSize_0: [X₁-1-X₂]
• eval_noiseTickSize_1: [X₁-1-X₂]
• eval_noiseTickSize_bb1_in: [X₁-1-X₂]

MPRF for transition t₂: eval_noiseTickSize_bb1_in(X₀, X₁, X₂, X₃) → eval_noiseTickSize_0(X₀, X₁, X₂, X₃) :|: X₃ ≤ 10001+X₂ ∧ X₃ ≤ 10001 ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

20006⋅X₁+30008 {O(n)}

MPRF:

• eval_noiseTickSize_0: [10001-X₃]
• eval_noiseTickSize_1: [10001-X₃]
• eval_noiseTickSize_bb1_in: [10002-X₃]

MPRF for transition t₂₃: eval_noiseTickSize_1(X₀, X₁, X₂, X₃) → eval_noiseTickSize_bb1_in(X₀, X₁, X₂, 1+X₃) :|: X₃ ≤ 10000 ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

20004⋅X₁+30005 {O(n)}

MPRF:

• eval_noiseTickSize_0: [10001-X₃]
• eval_noiseTickSize_1: [10001-X₃]
• eval_noiseTickSize_bb1_in: [10001-X₃]

knowledge_propagation leads to new time bound 20006⋅X₁+30008 {O(n)} for transition t₄: eval_noiseTickSize_0(X₀, X₁, X₂, X₃) → eval_noiseTickSize_1(nondef.0, X₁, X₂, X₃) :|: 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃

All Bounds

Timebounds

Overall timebound:60018⋅X₁+90030 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 20006⋅X₁+30008 {O(n)}
t₄: 20006⋅X₁+30008 {O(n)}
t₅: X₁+1 {O(n)}
t₈: X₁+1 {O(n)}
t₉: 1 {O(1)}
t₁₂: 1 {O(1)}
t₂₃: 20004⋅X₁+30005 {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}
t₂₆: 1 {O(1)}

Costbounds

Overall costbound: 60018⋅X₁+90030 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 20006⋅X₁+30008 {O(n)}
t₄: 20006⋅X₁+30008 {O(n)}
t₅: X₁+1 {O(n)}
t₈: X₁+1 {O(n)}
t₉: 1 {O(1)}
t₁₂: 1 {O(1)}
t₂₃: 20004⋅X₁+30005 {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}
t₂₆: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: 0 {O(1)}
t₁, X₃: 0 {O(1)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: 2⋅X₁+2 {O(n)}
t₂, X₃: 20004⋅X₁+30007 {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: 2⋅X₁+2 {O(n)}
t₄, X₃: 20004⋅X₁+30007 {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: 2⋅X₁+2 {O(n)}
t₅, X₃: 1 {O(1)}
t₈, X₁: X₁ {O(n)}
t₈, X₂: 2⋅X₁+2 {O(n)}
t₈, X₃: 1 {O(1)}
t₉, X₁: X₁ {O(n)}
t₉, X₂: 2⋅X₁+2 {O(n)}
t₉, X₃: 20004⋅X₁+30007 {O(n)}
t₁₂, X₁: X₁ {O(n)}
t₁₂, X₂: 2⋅X₁+2 {O(n)}
t₁₂, X₃: 20004⋅X₁+30007 {O(n)}
t₂₃, X₀: 0 {O(1)}
t₂₃, X₁: X₁ {O(n)}
t₂₃, X₂: 2⋅X₁+2 {O(n)}
t₂₃, X₃: 20004⋅X₁+30007 {O(n)}
t₂₄, X₀: 0 {O(1)}
t₂₄, X₁: X₁ {O(n)}
t₂₄, X₂: 2⋅X₁+2 {O(n)}
t₂₄, X₃: 20004⋅X₁+30007 {O(n)}
t₂₅, X₀: 0 {O(1)}
t₂₅, X₁: X₁ {O(n)}
t₂₅, X₂: 2⋅X₁+2 {O(n)}
t₂₅, X₃: 20004⋅X₁+30007 {O(n)}
t₂₆, X₁: 4⋅X₁ {O(n)}
t₂₆, X₂: 8⋅X₁+8 {O(n)}
t₂₆, X₃: 80016⋅X₁+120028 {O(n)}